monotone

Examples of monotone in the following topics:

• Alternating Series

  • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms $a_n$ converge to $0$ monotonically.
  • Proof: Suppose the sequence $a_n$ converges to $0$ and is monotone decreasing.
  • Since $a_n$ is monotonically decreasing, the terms are negative.
  • $a_n = \frac1n$ converges to 0 monotonically.

• The Mean Value Theorem, Rolle's Theorem, and Monotonicity

• The Integral Test and Estimates of Sums

  • Consider an integer $N$ and a non-negative function $f$ defined on the unbounded interval $[N, \infty )$, on which it is monotonically decreasing.
  • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:

• Tips for Testing Series

  • Integral test: For a positive, monotone decreasing function $f(x)$ such that $f(n)=a_n$, if $\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty$ then the series converges.